Selected Titles in This Series · Selected Titles in This Series 90 Christian Gerard and Izabella Laba, Multiparticle quantum scattering in constant magnetic fields, 2002 89 Michel

Selected Titles in This Series

90 Christian Gerard and Izabella Laba, Multiparticle quantum scattering in constant magnetic fields, 2002

89 Michel Ledoux, The concentration of measure phenomenon, 2001 88 Edward Frenkel and David Ben-Zvi , Vertex algebras and algebraic curves, 2001 87 Bruno Poizat , Stable groups, 2001 86 Stanley N . Burris, Number theoretic density and logical limit laws, 2001 85 V. A. Kozlov, V. G. Maz'ya, and J. Rossmann, Spectral problems associated with

corner singularities of solutions to elliptic equations, 2001 84 Laszlo Fuchs and Luigi Salce, Modules over non-Noetherian domains, 2001 83 Sigurdur Helgason, Groups and geometric analysis: Integral geometry, invariant

differential operators, and spherical functions, 2000 82 Goro Shimura, Arithmeticity in the theory of automorphic forms, 2000 81 Michael E. Taylor, Tools for PDE: Pseudodifferential operators, paradifferential

operators, and layer potentials, 2000 80 Lindsay N . Childs, Taming wild extensions: Hopf algebras and local Galois module

theory, 2000 79 Joseph A. Cima and Wil l iam T. Ross , The backward shift on the Hardy space, 2000 78 Boris A. Kupershmidt , KP or mKP: Noncommutative mathematics of Lagrangian,

Hamiltonian, and integrable systems, 2000 77 Fumio Hiai and Denes Petz , The semicircle law, free random variables and entropy,

2000 76 Frederick P. Gardiner and Nikola Lakic, Quasiconformal Teichmuller theory, 2000 75 Greg Hjorth, Classification and orbit equivalence relations, 2000 74 Daniel W . Stroock, An introduction to the analysis of paths on a Riemannian manifold,

2000 73 John Locker, Spectral theory of non-self-adjoint two-point differential operators, 2000 72 Gerald Teschl, Jacobi operators and completely integrable nonlinear lattices, 1999 71 Lajos Pukanszky, Characters of connected Lie groups, 1999 70 Carmen Chicone and Yuri Latushkin, Evolution semigroups in dynamical systems

and differential equations, 1999 69 C. T. C. Wall (A. A. Ranicki, Editor) , Surgery on compact manifolds, second edition,

1999 68 David A. Cox and Sheldon Katz , Mirror symmetry and algebraic geometry, 1999 67 A. Borel and N . Wallach, Continuous cohomology, discrete subgroups, and

representations of reductive groups, second edition, 2000 66 Yu. Ilyashenko and Weigu Li, Nonlocal bifurcations, 1999 65 Carl Faith, Rings and things and a fine array of twentieth century associative algebra,

1999 64 Rene A. Carmona and Boris Rozovskii , Editors, Stochastic partial differential

equations: Six perspectives, 1999 63 Mark Hovey, Model categories, 1999 62 Vladimir I. Bogachev, Gaussian measures, 1998 61 W . Norrie Everitt and Lawrence Markus, Boundary value problems and symplectic

algebra for ordinary differential and quasi-differential operators, 1999 60 Iain Raeburn and Dana P. Wil l iams, Morita equivalence and continuous-trace

C*-algebras, 1998 For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore / .

http://dx.doi.org/10.1090/surv/090

Multiparticle Quantum Scattering in Constant Magnetic Fields

Multiparticle Quantum Scattering in Constant Magnetic Fields

Christian Gerard

Izabella Laba

American Mathematical Society

Editorial Board Peter Landweber Tudor Ratiu Michael Loss, Chair J. T. Stafford

2000 Mathematics Subject Classification. Primary 35P25, 35Q40, 34L25, 47A40, 81U10.

ABSTRACT. This book is devoted to the scattering theory of systems of N interacting quantum particles in an external constant magnetic field. Particular emphasis is placed on the develop­ment of the Mourre theory, applications of geometrical methods, and the proof of asymptotic completeness for a large class of systems.

Library of Congress Cataloging-in-Publication D a t a Gerard, Christian, 1960-

Multiparticle quantum scattering in constant magnetic fields / Christian Gerard, Izabella Laba. p. cm. — (Mathematical surveys and monographs ; ISSN 0076-5376 ; v. 90)

Includes bibliographical references and index. ISBN 0-8218-2919-X (alk. paper) 1. Scattering (Physics) 2. Quantum theory. 3. Few-body problem. 4. Magnetic fields. I. Laba,

Izabella, 1966- II. Title. III. Series.

QC20.7.S3 G47 2001 539.7'58—dc21 2001053521

Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given.

Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Assistant to the Publisher, American Mathematical Society, P. O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to [email protected].

© 2002 by Christian Gerard and Izabella Laba. Printed in the United States of America.

@ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability.

Visit the AMS home page at URL: http:/ /www.ams.org/

10 9 8 7 6 5 4 3 2 1 06 05 04 03 02 01

Contents

Preface ix

Notation xi

Chapter 1. Fundamentals 1 1.1. Introduction 2 1.2. Hamiltonians 9 1.3. Reducing transformations 21 1.4. Bound and scattering states 36

Chapter 2. Geometrical Methods I 45 2.1. N-body geometry 46 2.2. Channel Hamiltonians 50 2.3. Partitions of unity 54

Chapter 3. The Mourre Theory 65 3.1. Local inequalities for operators 68 3.2. Basics of the Mourre theory 71 3.3. The strongly charged case 74 3.4. Analytically fibered operators 80 3.5. Two-particle neutral systems 84 3.6. Dispersive channels 94 3.7. Three-dimensional dispersive systems 104 3.8. Two-dimensional dispersive systems 141

Chapter 4. Basic Propagation Estimates 143 4.1. Preliminaries 144 4.2. Asymptotic energy and maximal velocity estimates 150 4.3. Asymptotic velocity along the field 152 4.4. Minimal velocity estimates 155

Chapter 5. Geometrical Methods II 167 5.1. The center of orbit observable 168 5.2. Admissible functions 172 5.3. Propagation estimates in the transversal direction 184

Chapter 6. Wave Operators and Scattering Theory 201 6.1. Introduction 201 6.2. Channel identification operators 205

viii CONTENTS

6.3. Three-dimensional charged systems 206 6.4. Three-dimensional dispersive systems 213 6.5. Two-dimensional charged systems 221 6.6. Two-dimensional systems with neutral pairs 223

Chapter 7. Open Problems 227

Chapter 8. Appendix 231

Bibliography 235

Index 241

Preface

This monograph is devoted to the spectral and scattering theory of quan­tum Hamiltonians describing systems of TV interacting particles in an exter­nal constant magnetic field. Most of it consists of the results obtained by the authors from 1993 to 1999.

Quantum scattering theory is the subfield of quantum mechanics which deals with the large-time asymptotics of the solutions of the Schrodinger equation and with the structure of the continuous spectrum of the corre­sponding Schrodinger operator. One of its main problems is to prove (or disprove) asymptotic completeness, which roughly speaking, is a statement that all solutions of the Schrodinger equation under consideration must fol­low asymptotically certain prescribed patterns. (The precise mathematical formulation of this will be given in Chapter 6.) There is a vast body of literature on this and other aspects of 2-particle scattering, see e.g., [RS] or [Ho, vols. II, IV] for an overview. For TV > 3 particles, the problem becomes much more complicated. It was only in the last 20 years that the TV-body scattering theory underwent a period of rapid development, begin­ning with the work of Enss [E2], [E3], and culminating in the proof of TV-body asymptotic completeness by Sigal-Soffer [SSI] and Derezihski [Del], with significant contributions by many other authors, see e.g., [Ml], [PSS], [FH1], [Gr], [Y]. We refer the reader to [DG1] for a more detailed account of that story and for a self-contained presentation of the results obtained in the 1980's and 90's.

Our work was largely inspired by these developments: we set out to extend the new results on asymptotic completeness to the case of TV-body systems in a constant magnetic field. Such systems are of considerable in­terest in quantum physics. There is a large body of research on the quantum Hall effect; most of it assumes that there are no interactions between the particles save for the Pauli exclusion principle, but it is possible that at some point the scattering effects will have to be taken into account. In as­trophysics, there is some evidence that strong magnetic fields exist on the surfaces of neutron stars and white dwarfs. "Quantum dots" are a prime example of quantum systems which can be significantly affected by magnetic fields of strength comparable to what can actually be achieved in existing laboratories. Physicists have also been studying highly excited (Rydberg) atoms in magnetic fields, which offer an opportunity to study the phenomena of "quantum chaos". See e.g., [RWHG] for a survey of some of the recent

ix

x PREFACE

(theoretical and experimental) work on the subject. Furthermore, there is a growing interest in magnetic Hamiltonians among mathematicians and mathematical physicists. In particular, questions such as the stability of matter [Li], [LSS], [Fe], eigenvalue or resonance asymptotics [Ivl], [Iv2], [FW1], [FW2], [FW3], and decay of eigenfunctions [Er], [N], [So] were recently addressed in the literature.

The purpose of this book is twofold. Firstly, in Chapter 1 we provide a general introduction to the spectral theory of TV-body magnetic Hamil­tonians, aimed at a wider audience of mathematical physicists. Secondly, we present a proof of asymptotic completeness for wide classes of magnetic Hamiltonians, namely for generic 3-body systems and for TV-body systems whose all proper subsystems have nonzero total electric charge. The proof requires much more than simply applying the known methods in a slightly different situation; this book focuses on the new methods and techniques that are specific to the magnetic case. In particular, this includes an ex­tension of the Mourre theory to "dispersive" Hamiltonians with a rather complicated structure (Chapter 3) and a geometrical analysis of the prop­agation of charged systems (Chapter 5). Our goal was to give a clear and reasonably self-contained presentation of the subject and to provide a solid foundation for further research.

The book is addressed mostly to researchers and graduate students in mathematical physics. We do expect the reader to be familiar with quan­tum mechanics, functional analysis, and modern PDE theory (especially with pseudodifferential calculus). A background in TV-body scattering and abstract Mourre theory will be useful, but not indispensable. To the readers who wish to acquire such background we recommend the monographs [DG1] and [ABG]. However, anyone willing to accept without proof the results of [DG1] and [ABG] that we will invoke should also be able to follow all of our arguments. In fact parts of this book (especially Chapters 1 and 2) may serve as an introduction to the TV-body theory. We emphasize that no previous exposure to magnetic Schrodinger operators is required.

Some of the results presented here were first published in [GL1-3]. How­ever much of the material, including all of our results in the 2—dimensional case and a large part of the geometrical analysis of Chapter 5, is published here for the first time. The Mourre theory for magnetic Hamiltonians (Chap­ter 3) has been completely reworked and rewritten, especially in the case we call "dispersive".

Notation

For the reader's convenience, we collect and explain here the notation used throughout this book.

Spaces: X, Y, Z,X, Y, Z, often subscripted or superscripted, will be Eu­clidean spaces (isomorphic to Rm for some m). The coordinates in these spaces will be denoted by x, y, z, x, y, z, with appropriate subscripts or superscripts. The Roman letters X, Y, Z, will be used to denote the con­figuration spaces before the center of mass separation, and the italic X, Y, Z denote the same spaces after the center of mass separation. (Most of the time we will only work with the latter spaces.) A similar convention will be used for the coordinates x, x, etc. Since we will actually separate the center of mass only in one direction, we will have Y = Y and y = y. Dual spaces: The spaces dual to X, Y, Z, X, Y, Z will be denoted by X', Y', etc. The duality will be denoted by (-,•). If X is one of the configuration spaces as above, T*X := X x Xr will be the cotangent bundle of X. Derivatives: The symbol dxf or -7^ will mean the partial derivative of f in x if x is a variable in R, and Vxf if x is a vector variable. We will also write Dxf = —idxf. We will often omit the subscripts x if the choice of variables is clear from context, and abbreviate dXj to <9j, DXj to D^ etc. For a — ( a i , . . . , a n ) , we use the standard notation

For 0 < e < 1, we will denote:

Ce(R) = {fe C(R) : \f(x + y)- f(y)\ < c(x)\y\e for all x G R, \y\ < 1},

Ck+€(R) = {fe Ck(R) : f{k) G Ce(R)}.

Classes of symbols: For an Euclidean space X (usually equal to one of the configuration spaces above), S{X) is the Schwartz class of functions on X, and S'(X) is the space of tempered distributions on X. We will also work with the following classes of symbols:

S\X) = {f€ C°°{X)\ \daf{x)\ <ca, xe X, \a\ > 0},

and, for e G R,

Secl{X) = {fe C°°(X)\ \Daf(x)\ < cQ(*>£- |a |, xeX, \a\ > 0}.

xi

xii NOTATION

We will use the convention that whenever an estimate is stated for all func­tions / i , /2, • • • in a specified class of symbols 5, the constants in the estimate are understood to depend only on the seminorms of fi in S, and that this dependence will be linear for each fi.

Opera tors : H, H!\ etc., often subscripted or superscripted, will be Hilbert spaces, usually equal to I?(X), where X is one of the configuration spaces above. The inner product on H will be denoted by (•,•), and will be linear in the second variable and antilinear in the first one. A, B, iJ, etc. will be linear operators on H. For Hamiltonians of the iV-body systems under consideration, we adopt the convention that the Roman H and the italic H are, respectively, the Hamiltonians of the system before and after the center of mass separation. The identity operator on H will be denoted by 1* ,̂ or simply by 1 if it is clear from context what H is. B(Hi,0,2) is the algebra of bounded linear operators from Hi to H2', we will abbreviate B(H,H) =: B(H). We will use s— lim and w— lim to denote the limits in the strong and weak topology, respectively. T>(H) is the domain of H; it will always be either specified explicitly or clear from context whether the form domain or the operator domain of H is meant. The graph norm on T>(H) is

\W\\v(H) = IMI + \\Hu\\.

The symbols cr(i?), crpp(i7), crcont(H) will denote the spectrum of H, the point spectrum of i7, and the continuous spectrum of if, respectively. We will also use HPP(H), HCOnt(H), W a c(#) ^° denote the pure point, continu­ous, and absolutely continuous spectral subspaces of H. 1Q(H) is the spectral projection of the self-adjoint operator if on a set tt cK. If A is a symmetric quadratic form and B is a symmetric operator on 7Y, the phrase UA is bounded on the domain of B" means that

\A{u,u)\ <C||£7i||2, ueV(B). (In particular, it implies the inclusion T>(A) C V(B2) between the form domains of A and B2.) If A is an operator, the phrase UA preserves the domain of B" means that Hue V{B) H V(A), then An e V(B) and

\\BAu\\ <C(\\u\\ + \\Bu\\).

A mapping M 3 k —> A(k), where M i s a metric space (usually a subset of Rn) and A(k) are self-adjoint operators on 7Y, is said to be continuous in the norm resolvent sense if the mapping

M3k^(A(k) + z)~1 eB(H)

NOTATION xiii

is norm continuous in k for any z with I m z ^ 0, and analytic in the norm resolvent sense if the above mapping is analytic in k for any z with I m z ^ O . We will use UA + h.c." to denote UA + A*; typically, this notation will be employed when A is a long expression of the form A — A\ . . . Ak, in which case the "h.c." stands for A*k . . . A*.

We define sdA(H) := [H,iA] and a d ^ ( # ) := [ a d ^ t f ) , A]. By "undoing a commutator" we will mean writing [A, B] as AB — BA and estimating each term separately. If A(t) is a family of operators on H depending on a parameter t, we will write A(t) = 0{ta) if A{t) G B(H) for t large enough and ||A(t)| | = 0{ta). A similar convention will be used for A(t) = o(ta). The notation "A>*B at H — A", used heavily in Chapter 3, is explained at the beginning of Section 3.1. The classes of operators CS(A)1 where A is a self-adjoint operator on 71, are defined in Section 3.2. Cut-off functions: 1Q will be the characteristic function of the set ft. F(x G £1) is a "smoothed out" characteristic function of Q\ it is a C°° function equal to 0 outside O and to 1 in a slightly smaller set. Misce l lanous: (x) is a function in C°°(X), greater than 1/2 for all x G X and equal to \x\ for \x\ > 1.

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Index

a-compact operators, 75

admissible function, 168, 170, 174 admissible function, construction of, 182 Agmon Hamiltonians, 46, 227 algebraic scattering theory, 202 almost analytic extension, 19 almost positive operators, 68 analytically decomposable operator, 80 asymptotic completeness, 4, 8, 9, 201 asymptotic energy, 150 asymptotic observable, 147, 203 asymptotic velocity, 5, 9, 143, 152, 203

bound states, 4, 6, 8, 21, 36

center of charge, 7, 173 center of charge coordinates, 170, 173 center of mass, 5, 21, 48 center of mass coordinates, 23 center of mass separation, 36 center of mass, velocity of, 42 center of orbit, 3, 7, 9, 167, 168 chain rule, 147 channel Hamiltonian, 45, 50 channel identification operator, 205 channel reducing transfonmation, 52 cluster, 8, 45, 46 cluster decomposition, 45, 47 cluster decomposition, charged, 49 cluster decomposition, refinement of, 47 cluster decomposition, strongly charged,

49 cluster, charged, 47 cluster, Hamiltonian of, 50 cluster, neutral, 47 comparison operator, 17 conjugate operator, 65, 73 Cook's method, 146 Coulomb potential, 10, 14, 37 Coulomb-type potential, 13, 220, 228

dilations, generator of, 77, 89

direct integral, 7, 20, 27, 41, 80 dispersive Hamiltonians, 37, 65, 229 Dollard dynamics, 202 Dollard modifiers, 211

exponential decay of eigenfunctions, 86, 230

Garding inequality, sharp, 18

harmonic oscillator, 33, 38, 91, 106

intercluster interaction, 51

Kato's inequality, 14

Landau levels, 3, 34 Leibniz rule, 73 long-range potentials, 5, 202, 211

magnetic Sobolev space, 14 magnetic translations, 3, 25 maximal velocity estimate, 143, 150 minimal velocity estimate, 65, 144, 155 modified free evolution, 202 Mourre estimate, 9, 65, 73 Mourre estimate, strict, 74 multichannel scattering, 45 Murray-von Neumann's theorem, 28

Nelson's commutator theorem, 14, 16

particle, charged, 2 particle, neutral, 2, 229 partition of unity, conical, 57 partition of unity, cylindrical, 54 partition of unity, N-body, 54 Pauli Hamiltonian, 10 Poisson bracket, 32 potentials, assumptions on, 13 propagation estimates, 143 pseudomomentum, 3, 25 pseudomomentum of the center of mass,

25 pseudomomentum, functions of, 29

242 INDEX

pseudomomentum, total, 6, 22 Putnam-Kato theorem, 149

real interpolation spaces, 130 real interpolation theorem, 131 reduced Hamiltonian, 14, 21, 22, 28, 29 reducing transformation, 26, 28 regularly generated dynamics, 145

scattering channel, 8 scattering cross-sections, 203 scattering operator, 4, 202 scattering states, 4, 6, 8, 36 short-range potentials, 4, 5, 201 symplectic transformation, 18, 26, 28, 52

threshold set, 75, 80, 94, 105 threshold set, secondary, 96, 98

time-dependent Hamiltonian, 144 time-dependent observable, 147 transformed Hamiltonian, 27, 53 transversal gauge, 2, 10

unitary dynamics, 144

vector potential, 9 vector potential, TV-particle, 11, 12

wave operators, 4, 146, 201 wave operators, channel, 8, 202 wave operators, modified, 5, 211 wave operators, short-range, 209 Weyl quantization, 18 Weyl symbol, 18